Question

$\cos \theta -\sin \theta =\sqrt{2}\sin \theta$ show that $\cos \theta +\sin \theta =\sqrt{2}\cos \theta$

Answer 1

$\cos \theta -\sin \theta =\sqrt{2}\sin \theta$

${(\cos \theta -\sin \theta )}^2=2{\sin }^2\theta$

${\cos }^2\theta +{\sin }^2\theta -2\sin \theta \cos \theta =2{\sin }^2\theta$

${\cos }^2\theta -{\sin }^2\theta -2\sin \theta \cos \theta =0$

${\cos }^2\theta -{\sin }^2\theta -2\sin \theta \cos \theta -2{\cos }^2\theta =-2{\cos }^2\theta$

$-{\cos }^2\theta -{\sin }^2\theta -2\sin \theta \cos \theta =-2{\cos }^2\theta$

${\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta =2{\cos }^2\theta$

${(\sin \theta +\cos \theta )}^2=2{\cos }^2\theta$

$\sin \theta +\cos \theta =\sqrt{2}\cos \theta$